On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69–115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke’s conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection.     

Invariant tensorfields and the canonical connection of a 3-web

In the following we shall give a new approach to web geometry. Instead of using the reduced adapted coframe bundles over the web manifolds we define invariant tensorfields corresponding to the 3-web structure and express the covariant derivation of the Chern connection with the help of these tensorfields, working only on the tangent bundle of the web manifold. The invariant tensorfields of a 3-web define a more general, so-called {H, J}-structure on the manifold which can be considered as an infinitesimal, non-integrable version of a web structure. We introduce a canonical connection of a {H, J}-structure which reduces to the Chern connection in the case of a 3-web. Using the tensorial expression of the covariant derivation of the Chern connection we give direct proof for the torsion and curvature identities. Finally, we apply our formulae to an algebraic characterization of the parallel translation with respect to the Chern connection.     

The category of 3-computads is not cartesian closed

We show, using [A. Carboni, P.T. Johnstone, Connected limits, familial representability and Artin glueing, Math. Structures Comput. Sci. 5 (1995) 441-459] and Eckmann-Hilton argument, that the category of 3-computads is not cartesian closed. As a corollary we get that neither the category of all computads nor the category of n-computads, for n>2, do form locally cartesian closed categories, and hence elementary toposes.     

The \mathfrak {sl}_{3}-web algebra

In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$ -skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$ , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.     

On the multidimensional zeta functions associated with theta functions,and the multidimensional Appell polynomials

We define and study the multidimensional Appell polynomials associated with theta functions. For the trivial theta functions, we obtain the various well-known Appell polynomials. Many other interesting examples are given. To push our study, by Mellin transform, we introduce and investigate the multidimensional zeta functions associated with thetas functions and prove that the multidimensional Appell polynomials are special values at the nonpositive integers of these zeta functions. Using zeta functions techniques, among others, we prove an induction formula for multidimensional Appell polynomials. The last part of this paper is devoted to spectral zeta functions and its generalization associated with Laplacians on compact Riemannian manifolds. From this generalization, we construct new Appell polynomials associated with Riemannan manifolds of finite dimensions.     

Discretization of multidimensional submanifolds associated with spin-valued spectral problems

We present a large family of Spin(p, q)-valued discrete spectral problems. The associated discrete nets generated by the so-called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These nets are discrete analogues of smooth multidimensional immersions in ℝ^m including isothermic surfaces, Guichard nets, and some other families of orthogonal nets. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 253–262, 2006.     

One and multidimensional sampling theorems associated with Dirichlet problems

We use the eigenfunction expansion of Green's function of Dirichlet problems to obtain sampling theorems. The analytic properties of the sampled integral transforms as well as the uniform convergence of the sampling series are proved without any restrictions on the integral transforms. We obtain a one- and multi-dimensional versions of sampling theorems. In both cases the sampling series are written in terms of Lagrange-type interpolation expansions. Some examples and the truncation error as well as the stability of the obtained sampling expansions are discussed at the end of the paper. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On closures verifying that the interior of a closed element is closed

We give significant improvements to the Graham, Knuth & Motzkin result: if R is any binary relation, R ^+c+c+ = R ^+c+c (where P ⁺ denotes the transitive closure of the binary relation P, and P ^c its Boolean complement)     

Optimal control of a functional equation associated with closed range selfadjoint operators

Necessary and sufficient conditions for the optimality of a pair subject to are given. Here is a selfadjoint operator with closed range on a Hilbert space and . The case - unbounded is also discussed, which leads to some open problems. This general functional scheme includes most of the previous results on the optimal control of the -periodic wave equation for all in a dense subset of . It also includes optimal control problems for some elliptic equations.

     

The brown-MCcoy radical is not closed under products

In this paper we answer the question raised by Fisher whether the Brown-McCoy radical is closed under products or equivalently - under ultraproducts. Morever, we prove that the radical is closed under ultraproducts with respect ω-complete ultrafilters.     

Maximal sectorial extensions and closed forms associated with them

We describe all closed sesquilinear forms associated with m-sectorial extensions of a densely defined sectorial operator with vertex at the origin.     

The smallest ideal of βS is not closed

Let S be an infinite discrete semigroup which can be embedded algebraically into a compact topological group and let βS be the Stone–Čech compactification of S. We show that the smallest ideal of βS is not closed.     

The radical of a Banach algebra with a closed cone

LetA be a real Banach algebra with a closed cone. If the cone contains all squares then the radical only consists of elementsx εA withx ³=0. This characterization yields an elementary proof of a result of Scheffold forff-Banach lattice algebras.